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Chapter 3: Problem 26

Spreading Grass Seed. Emily seeds lawns for Sam's Superior Lawn Care. When she walks at a rapid pace, the wheel on the broadcast spreader completes \(150 \frac{2}{3}\) revolutions per minute. How many revolutions does the wheel complete in \(15 \mathrm{~min} ?\)

Short Answer

Expert verified
2260 revolutions

Step by step solution

01

Determine the revolutions per minute

Emily's broadcast spreader wheel completes 150 revolutions per minute plus 2/3 of a revolution. Let's express this as a mixed number: \(150 \frac{2}{3} = 150 + \frac{2}{3}\).
02

Convert mixed number to improper fraction

First, convert 150 \(\frac{2}{3}\) to an improper fraction: \(150 \frac{2}{3} = 150 + \frac{2}{3} = \frac{450}{3} + \frac{2}{3} = \frac{452}{3}\).
03

Calculate total revolutions in 15 minutes

The wheel completes \(\frac{452}{3}\) revolutions per minute. In order to find the total number of revolutions in 15 minutes, multiply this amount by 15: \[ \text{Total Revolutions} = 15 \times \frac{452}{3} \]
04

Simplify multiplication

To simplify, calculate \[ = \frac{15 \times 452}{3} \]Solve the multiplication in the numerator first:\[ 15 \times 452 = 6780 \]Then divide the product by the denominator:\[ \frac{6780}{3} = 2260 \]
05

Present the final answer

The total number of revolutions the wheel completes in 15 minutes is 2260.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mixed Number Conversion
A mixed number combines a whole number and a fraction. In this exercise, we start with the mixed number \(150 \frac{2}{3}\). To handle mixed numbers in calculations, it's easier to convert them into improper fractions. An improper fraction has a numerator larger than its denominator.
To convert \(150 \frac{2}{3}\), follow these steps:
  • Multiply the whole number by the denominator of the fraction: \(150 \times 3 = 450\).

  • Add the numerator of the fraction to this product: \(450 + 2 = 452\).

  • The result is the numerator of the improper fraction, with the same denominator: \(\frac{452}{3}\).
This improper fraction is more straightforward to use in multiplication and division.
Improper Fraction
An improper fraction is useful when performing multiplication or division in calculations. Converting \(150 \frac{2}{3}\) to the improper fraction \(\frac{452}{3}\) simplifies the problem significantly.
Recap of creating improper fractions:
  • For a mixed number like \(a \frac{b}{c}\), compute \(a \times c + b\); this is the new numerator.

  • Keep the denominator the same: \(\frac{a \times c + b}{c}\).
This form allows you to perform arithmetic operations directly, without managing whole numbers and fractions separately.
Multiplication
Multiplying fractions and whole numbers requires a simple adjustment. To find the total revolutions for 15 minutes, we multiply the improper fraction by 15. Here's how:
  • Express the multiplication: \(15 \times \frac{452}{3}\).

  • Transform this into a fraction multiplication: \(\frac{15}{1} \times \frac{452}{3}\).
Multiplying fractions involves multiplying the numerators and denominators:
  • The new numerator: \(15 \times 452 = 6780\).

  • The new denominator: \(1 \times 3 = 3\).
This results in \(\frac{6780}{3}\), which simplifies easily to provide the final calculation.
Simplification
Simplifying fractions turns complex fractions into simple, understandable numbers. Here, we simplify \(\frac{6780}{3}\) by performing the division:
  • First, compute the division: \(6780 \div 3 = 2260\).
This simplification transforms the fraction into a whole number, providing an easy-to-understand result.
In summary, the wheel completes **2260 revolutions** in 15 minutes.

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